Rudy Giuliani stakes his presidential bid on his record of cutting crime in New York – but the union representing the city’s 30,000 police officers won’t support his run for the White House.

“The New York City Patrolmen’s Benevolent Association could never support Rudy Giuliani for any elected office,” PBA President Patrick Lynch told The Post.

Lynch also claimed that “Rudy Giuliani has no real credentials as a terrorism fighter.”

The PBA leader’s dismissal of Giuliani’s candidacy reflects a paradox in the law-and-order ex-mayor’s eight-year tenure at City Hall: The man who prides himself on having driven down crime and made New York “the safest big city in America” is unpopular with many of the cops he once led.

The rocky relationship stems largely from years of labor disputes over how much to pay cops. In the first two years of a five-year contract under Giuliani, officers were given a salary freeze.

PBA members called it “zeroes for heroes” even though the package raised salaries 13.3 percent over the final years of the five-year pact.

Lynch said cops haven’t forgotten.

“The inability to keep veteran cops on the job or to recruit adequate numbers of new ones can be traced directly back to the Giuliani mayoralty,” Lynch said.

“While the city was rolling in money, the Giuliani administration cried future poverty and stuck New York City police officers with 3½ years without a raise. Giuliani’s ‘zeroes for heroes’ contracts held police pay stagnant while all the other local departments in the metro area were getting modest but steady raises.

“Today,” Lynch went on, “there are simply not enough NYPD police officers to keep this city safe, and it is Giuliani’s fault.”

The Giuliani campaign yesterday said Lynch’s shot was off the mark.

“Mayor Giuliani has always had, and continues to have, strong support from law enforcement,” said a campaign spokeswoman, Maria Comella.

“Rudy has long supported New York’s Police Department and worked together [with the NYPD] to cut the city’s crime rate in half.”

Former Staten Island Borough President Guy Molinari, a co-chairman of Giuliani’s campaign in New York, said the ex-mayor gave out the best raises the city could afford at the time.

“You could only do what the budget permits you to do,” he said.

Molinari added that most cops voted for Giuliani as mayor.

Lynch also questioned Giuliani’s reputation as “America’s Mayor” in the aftermath of the Sept. 11 terror attacks.

“Giuliani has wrapped himself firmly in the cloak of 9/11 for his own political purposes. But the real heroes of 9/11 – those who helped to evacuate those towers and lived to tell the tale, and all those who participated in the recovery and cleanup – know the truth,” he said.